3.90 \(\int \frac{1}{\sqrt{3-3 x^2+2 x^4}} \, dx\)

Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-3 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{8} \left (4+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-3 x^2+3}} \]

[Out]

((3 + Sqrt[6]*x^2)*Sqrt[(3 - 3*x^2 + 2*x^4)/(3 + Sqrt[6]*x^2)^2]*EllipticF[2*Arc
Tan[(2/3)^(1/4)*x], (4 + Sqrt[6])/8])/(2*6^(1/4)*Sqrt[3 - 3*x^2 + 2*x^4])

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Rubi [A]  time = 0.0519937, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-3 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{8} \left (4+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-3 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3 - 3*x^2 + 2*x^4],x]

[Out]

((3 + Sqrt[6]*x^2)*Sqrt[(3 - 3*x^2 + 2*x^4)/(3 + Sqrt[6]*x^2)^2]*EllipticF[2*Arc
Tan[(2/3)^(1/4)*x], (4 + Sqrt[6])/8])/(2*6^(1/4)*Sqrt[3 - 3*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 4.50281, size = 88, normalized size = 0.98 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{2 x^{4} - 3 x^{2} + 3}{\left (\frac{\sqrt{6} x^{2}}{3} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{3} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | \frac{\sqrt{6}}{8} + \frac{1}{2}\right )}{12 \sqrt{2 x^{4} - 3 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x**4-3*x**2+3)**(1/2),x)

[Out]

6**(3/4)*sqrt((2*x**4 - 3*x**2 + 3)/(sqrt(6)*x**2/3 + 1)**2)*(sqrt(6)*x**2/3 + 1
)*elliptic_f(2*atan(2**(1/4)*3**(3/4)*x/3), sqrt(6)/8 + 1/2)/(12*sqrt(2*x**4 - 3
*x**2 + 3))

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Mathematica [C]  time = 0.174557, size = 142, normalized size = 1.58 \[ -\frac{i \sqrt{1-\frac{4 x^2}{3-i \sqrt{15}}} \sqrt{1-\frac{4 x^2}{3+i \sqrt{15}}} F\left (i \sinh ^{-1}\left (2 \sqrt{-\frac{1}{3-i \sqrt{15}}} x\right )|\frac{3-i \sqrt{15}}{3+i \sqrt{15}}\right )}{2 \sqrt{-\frac{1}{3-i \sqrt{15}}} \sqrt{2 x^4-3 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[3 - 3*x^2 + 2*x^4],x]

[Out]

((-I/2)*Sqrt[1 - (4*x^2)/(3 - I*Sqrt[15])]*Sqrt[1 - (4*x^2)/(3 + I*Sqrt[15])]*El
lipticF[I*ArcSinh[2*Sqrt[-(3 - I*Sqrt[15])^(-1)]*x], (3 - I*Sqrt[15])/(3 + I*Sqr
t[15])])/(Sqrt[-(3 - I*Sqrt[15])^(-1)]*Sqrt[3 - 3*x^2 + 2*x^4])

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Maple [C]  time = 0.119, size = 87, normalized size = 1. \[ 6\,{\frac{\sqrt{1- \left ( 1/2+i/6\sqrt{15} \right ){x}^{2}}\sqrt{1- \left ( 1/2-i/6\sqrt{15} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{18+6\,i\sqrt{15}},1/2\,\sqrt{-1-i\sqrt{15}} \right ) }{\sqrt{18+6\,i\sqrt{15}}\sqrt{2\,{x}^{4}-3\,{x}^{2}+3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4-3*x^2+3)^(1/2),x)

[Out]

6/(18+6*I*15^(1/2))^(1/2)*(1-(1/2+1/6*I*15^(1/2))*x^2)^(1/2)*(1-(1/2-1/6*I*15^(1
/2))*x^2)^(1/2)/(2*x^4-3*x^2+3)^(1/2)*EllipticF(1/6*x*(18+6*I*15^(1/2))^(1/2),1/
2*(-1-I*15^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} - 3 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 - 3*x^2 + 3),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 - 3*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} - 3 \, x^{2} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 - 3*x^2 + 3),x, algorithm="fricas")

[Out]

integral(1/sqrt(2*x^4 - 3*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{4} - 3 x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*x**4-3*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(2*x**4 - 3*x**2 + 3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} - 3 \, x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(2*x^4 - 3*x^2 + 3),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 - 3*x^2 + 3), x)